US7359477B2 - Method for reconstructing a CT image using an algorithm for a short-scan circle combined with various lines - Google Patents
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- A61B6/02—Devices for diagnosis sequentially in different planes; Stereoscopic radiation diagnosis
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- A61B6/44—Constructional features of apparatus for radiation diagnosis
- A61B6/4429—Constructional features of apparatus for radiation diagnosis related to the mounting of source units and detector units
- A61B6/4435—Constructional features of apparatus for radiation diagnosis related to the mounting of source units and detector units the source unit and the detector unit being coupled by a rigid structure
- A61B6/4441—Constructional features of apparatus for radiation diagnosis related to the mounting of source units and detector units the source unit and the detector unit being coupled by a rigid structure the rigid structure being a C-arm or U-arm
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- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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Definitions
- the present invention is directed to a method for reconstructing a CT image, and in particular to reconstructing a CT image using a reconstruction algorithm for a short-scan circle combined with various lines.
- the line scan can be regarded as an add-on to the conventional short-scan circular path.
- the approach should be theoretically exact, possess efficient, shift-invariant filtered back-projection (FBP) structure, and solve the long object problem.
- the algorithm should be flexible in dealing with various circle and line configurations.
- the reconstruction method should require nothing more than the theoretically minimum length of scan trajectory.
- a method for reconstructing a CT image of a subject from data acquired from the subject with a C-arm apparatus having an x-ray source with a focus from which x-rays emanate in a cone beam, and a radiation detector, mounted on a C-arm, by rotating the focus of the x-ray source around the subject through a focus trajectory and detecting radiation attenuated by the subject with the radiation detector.
- the C-arm is operated to move the focus of the x-ray source through an actual focus trajectory consisting of an actual incomplete circle and an actual straight-line segment attached at an end of the actual incomplete circle, and detecting radiation attenuated by the subject for each focus position in the actual focus trajectory.
- a projection matrix is electronically calculated that, for that focus position, describes a perspective cone beam projection of the subject on the radiation detector.
- An image of the subject is reconstructed using a known reconstruction algorithm that is based on an ideal focus trajectory consisting of an ideal incomplete circle and an ideal straight-line segment attached at an end of the ideal incomplete circle, with the ideal trajectory in the known reconstruction algorithm being adapted to the actual trajectory of the C-arm apparatus using the projection matrices.
- the inventive reconstruction algorithm is based on a reconstruction algorithm that uses an ideal source trajectory known from A. Katsevich, “Image reconstruction for the circle and line trajectory,” Phys. Med. Biol. 49, pp. 5059-5072, 2004 (from which the discussion below regarding the known inversion algorithm, and FIGS. 1-4 , are taken) and A. Katsevich, “A general scheme for constructing inversion algorithms for cone beam CT,” International Journal of Mathematics and Mathematical Sciences 21, pp. 1305-1321, 2003.
- C-arm devices exhibit certain mechanical instabilities that have to be considered. Fortunately, the geometrical deviations from the ideal source path are almost reproducible and are accounted for by a geometrical calibration process.
- the projection geometry of non-ideal source trajectories is described conveniently in the framework of projection matrices.
- the general use of projection matrices for describing projection geometry is discussed in R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision , Cambridge University Press, 2000.
- the back-projection step is performed exactly by a direct use of projection matrices.
- the filtering step requires a more elaborate adaption strategy.
- the inventive method is a simple but robust scheme to adapt the reconstruction algorithm to non-ideal sampling patterns as they occur in imaging with real world C-arm devices.
- FIG. 1 illustrates the basic circle and line trajectory for use in explaining the inventive method.
- FIG. 2 illustrates the projection onto the detector plane when the source is on the line.
- FIG. 3 illustrates the projection onto the detector plane when the source is on the circle.
- FIGS. 4 a and 4 b respectively illustrate examples of trajectories that can be handled using the inventive method.
- FIG. 5 illustrates the fit of an ideal circle and line trajectory into the set of real world (actual) focus positions, in accordance with the inventive method.
- FIG. 1 illustrates the circle and line trajectory.
- S2 is the unit sphere in IR3
- I I s ⁇ y(s) ⁇ L and I 2 s ⁇ y(s) ⁇ C are parameterizations of the line and circle, respectively. It is assumed that the circle is of radius R and centered at the origin.
- U be an open set, such that U ⁇ (x 1 ,x 2 ,x 3 ) ⁇ IR 3 :x 1 2 +x 2 2 ⁇ R 2 ⁇ .
- n The role of n is twofold. First, it has to deal with redundancy in the cone beam data by assigning weights to IPs between Radon planes and the source trajectory. Second, a proper choice of n yields an efficient shift-invariant convolution back-projection algorithm in the framework of Katsevich's general inversion formula.
- the function n described by Table 1, can be described as follows. If there is one IP, it is given weight 1. If there are three IPs, the two IPs on the circle have weight 1 each, and the IP on the line segment has weight ⁇ 1. As is easily seen, n is normalized:
- ⁇ is a polar angle in the plane perpendicular to ⁇ (s,x).
- u 2 ⁇ ( s , x ) ⁇ : y . ⁇ ( s ) ⁇ ⁇ ⁇ ( s , x ) ⁇ y . ⁇ ( s ) ⁇ ⁇ ⁇ ⁇ ⁇ ( s , x ) ⁇ , x ⁇ U , s ⁇ I 2 ⁇ ( x ) . ( 4 )
- u 2 (s,x) is the unit vector perpendicular to the plane containing x, y(s), and tangent to C at y(s).
- FIG. 2 illustrates the projection onto the detector plane when the source is on the line.
- FIG. 3 illustrates the projection onto the detector plane when the source is on the circle.
- the part of the support of f that can be accurately reconstructed by the algorithm can be determined.
- This is the volume bounded by the following three surfaces: the plane of C, the plane defined by L and the endpoint of C not on L, and the conical surface of lines joining the points of C to the endpoint of L that is not on C.
- This volume will be denoted U(C, L). It should be noted, however, that the object f may extend outside U(C, L), as long as it stays away from the source trajectory C ⁇ L.
- the trajectory consisting of an incomplete circle and a line segment can be used as a building block for constructing other trajectories.
- an incomplete circle C with line segments attached to it at each endpoint of C. These segments can be on opposite sides of C (see FIG. 4 ), or on the same side of C (see FIG. 4 b ).
- Inversion algorithms for these trajectories are obtained from (5) by applying it to each circle+line subset and then adding the results (if necessary). Indeed, suppose the segments are on opposite sides of C. Then the volume U(C, L) in the half-space z ⁇ 0 is reconstructed using the trajectory C ⁇ L, and the volume U(C, L′) in z ⁇ 0 is reconstructed using C ⁇ L′.
- d 1 and d 2 are the horizontal and vertical axes on the detector. Katsevich has shown that the line scans require the detector of size
- the non-ideal acquisition geometry of a real world C arm device is represented in the inventive method by a sequence of homogenous projection matrices P s ⁇ IR 3 ⁇ 4 .
- the matrix P s For every source position s, and thus for every measured projection image, the matrix P s completely describes the perspective cone beam projection of the object.
- u pix and are the coordinate values of an image point measured along the two perpendicular axes' vectors e u,s respectively e v,s that coincide with the row or the column direction of the pixel grid of the detector DP(s).
- an essential task is to determine an individual, valid sequence of P s for a given C-arm system.
- This geometric calibration is done by an automated procedure involving a calibration phantom of exactly defined structure and an appropriate calibration algorithm that calculates a valid matrix P s for a given s.
- a matrix P s can be decomposed in a complete set of projection parameters. Especially the extrinsic parameters are of interest as they include position and orientation of the involved detector and focus entities.
- the inventive method calculates the focus positions and the direction vectors of the pixel coordinate system's axes from every P s .
- the acquisition trajectory can be composed and when using matrices downloaded from a real world C-arm device, it is possible to determine the deviations of the acquisition geometry compared to the ideal circle and line geometry presumed by the reconstruction approach.
- the matrices M s ⁇ IR 3 ⁇ 3 are introduced, consisting of the first three columns of the P s . Note that all M s are invertible.
- the projection matrices define the detector only up to scale. To have knowledge about the precise structure of the C-arm acquisition system, either the specification of the focus-detector distance or the detector pixel spacing is needed. The direction of the two axis of the detector pixel coordinate system, however, is universally valid. e u,s is parallel to the vector ((0,0,1) ⁇ Ms) T ⁇ ((1,0,0) ⁇ M s ) T and e v,s points in the direction ((0,0,1) ⁇ Ms) T ⁇ ((0,1,0) ⁇ M s ) T .
- the projection matrix P s exactly describes the relation between the object and its cone beam projection image for every s.
- an exact consideration of the non-ideal acquisition geometry in the back-projection step is possible by the direct use of the projection matrices.
- the filtering step the case of non-ideal trajectory is transferred into the ideal case approximately.
- the presumed ideal trajectory consisting of a partial circle and a perpendicularly attached line segment is fitted into the set of real world focus positions.
- the fitted circle path again projects as a parabola onto the detector and the same approach as described above can be used to determine the filtering directions as tangents to the occurring parabola.
- FIG. 5 illustrates the fit of an ideal circle and line trajectory y fitted (s) into the set of real world focus positions y(s).
- a dotted source position is located below the circular plane CP.
- any appropriate cost function can be used to fit the ideal trajectory y fitted (s) into the path y(s).
- a least-square fit is described.
- the least-square fit of the ideal trajectory y fitted (S) into the path y(s), as illustrated in FIG. 5 corresponds to the minimization of the total estimation error
- ⁇ ⁇ s ⁇ ( ⁇ y fitted ⁇ ( s ) - y ⁇ ( s ) ⁇ 2 ) ( 12 ) and is done in a three steps approach.
- a least-square algebraic fit of a plane into the circle's focus positions is performed followed by an orthogonal projection of the path y(s) onto the determined circular plane CP.
- a partial circle is fitted into the projected focus positions using a 2D algebraic least-square estimation method and then optimally represents the circle scan.
- the line segment is determined perpendicular to the circular plane and connected to the end of the circle segment.
- the fitted trajectory can be described by the circular plane CP, the circle center x center , the circle segment's radius R and the length and the position of the line segment.
- a normalization of the coordinate system is performed. This is done by multiplying every P s from the right side with a transformation matrix T v ⁇ IR 4 ⁇ 4 independent from s.
- the normalization consists of two operations, a translation T v,t to locate the origin at x center and a rotation T v,r that parallelizes the line direction with the x 3 axis.
- T v T v,r ⁇ T v,t .
- the change of the relative position of the focus and the detector has to be handled.
- the detector coordinate system is adapted such that the fitted source trajectory is projected onto the detector on the same position as in the ideal case.
- the filtering instructions of the known inversion algorithm can be applied without any further modification.
- the used C-arm hardware it is sufficient to correct the in-plane translational movement of the pixel coordinate system, which is the most prominent deviation from the ideal geometry case.
- any other geometric deviation can be treated similarly.
- a translation matrix T d,s ⁇ IR 3 ⁇ 3 is determined and multiplied to P s from the left, so that the volume coordinate origin always projects onto the same detector coordinates. By that, the coordinates of ⁇ become independent from the current misplacement of the detector area.
- the object of interest in this experiment was composed of mathematically defined geometric objects like ellipsoids or cubes and simulates the basic anatomical structure of a human head including homogeneous regions with embedded low contrast objects but also high contrast structures. Because of its composition, this so-called mathematical head phantom (see, for example, “http://www.imp.uni-er Weg.de/phantoms/head/head.html, head phantom description) is very demanding to the reconstruction approaches significantly revealing any type of artifact.
- Severe cone artifacts appeared in the FDK reconstructed images in contrast to the high quality image data resulting from the circle and line method.
- the simulated scans of ideal and non-ideal trajectories started at different angular positions.
- the orientation of artifacts differed.
- additional artifacts in both reconstruction approaches were detected.
- Some streak-like artifacts of low intensity appeared near the high contrast bone structure.
- the artifacts were due to some slight irregularities in angular sampling and to some remaining inexactness in the filtering step.
- the matching of the contribution of the line and circular scan might be critical in more severe cases. Nevertheless, this experiment showed that the inventive adaptation is sufficient to consider geometrical distortions of real world C-arm devices.
Abstract
Description
It is assumed that f is smooth, compactly supported, and identically equals zero in a neighborhood of the source trajectory.
for almost all α∈S2. Here the summation is over all intersection points y(sj)∈II(x,α)∩Λ1π(x).
TABLE 1 |
Definition of the weight function n(s, x, α) |
Case | n | ||
1IP, s1 ε I1(x) | n(s1, x, α) = 1 | ||
1IP, s1 ε I2(x) | n(s1, x, α) = 1 | ||
3IPs, s1 ε I1(x) | n(s1, x, α) = −1 | ||
s2, s3 ε I2(x) | n(sk, x, α) = l, k = 2, 3 | ||
where θ is a polar angle in the plane perpendicular to β(s,x). According to the general scheme, described by Katsevich, jumps of φ(s,x,θ) have to be located in θ. By studying these jumps in two cases: s∈I1(x) and s∈I2(x) and using the general scheme the following inversion algorithm is obtained. Pick s∈I1(x) (i.e., y(s) is on the line). Find a plane through x and y(s), which is tangent to C at some yt(s,x), s∈I2(x). Let u1(s,x) be the unit vector perpendicular to that plane:
Pick now s∈I2(x) (i.e., y(s) is on the circle) and define
By construction, u2(s,x) is the unit vector perpendicular to the plane containing x, y(s), and tangent to C at y(s). Using (3) and (4) we obtain the following reconstruction formula for f∈C0 ∞(U):
where
Θk(s,x,γ):=cos γβ(s,x)+sin γe k(s,x),e k(s,x):=β(s,x)×u k(s,x). (6)
and δk is defined as follows:
δ1(s,x)==sgn(u 1(s,x)·{dot over (y)}(s)),s∈I 1(x); δ2(s,x)=1,s∈I 2(x). (7)
Suppose, for example, that L is parameterized in such a way that the source moves down along L as s increases. Then δ1(s, x)=1, s∈I1(x). If the source moves up along L as s increases, then δ1(s, x)=1, s∈I1(x).
depends on the region of interest (ROI) and is illustrated in
with 1/sin γ.
The circular scan thus requires a rectangular detector of a size
Here d1 and d2 are the horizontal and vertical axes on the detector. Katsevich has shown that the line scans require the detector of size
w h =P s x h, (10)
where a voxel with a Cartesian coordinate vector x is denoted by the homogenous vector xh−(b·xT,b) with b∈
y(s)=Ms −1 P s(0,0,0,1)T. (11)
and is done in a three steps approach.
describe the cone beam projection involving the normalized coordinate systems for the volume and for the detector. It should be noted that these projection matrices still represent the non-ideal acquisition geometry.
Experimental Results
TABLE 2 |
Reconstruction and Simulation Parameters |
mathematical head |
voxelized head | non-ideal | |||
ideal geometry | ideal geometry | geometry | ||
detector | 641 × 500 | 640 × 500 | 720 × 720 |
dimension | |||
pixel size | (0.6 mm)2 | (0.6 mm)2 | (0.580 mm)2 |
# projections | 501 | 501 | 538 |
on circle | |||
# projections | 221 | 196 | 196 |
on each line | |||
angular range | 200 deg | 200 deg | 214 deg |
of circle | |||
length of | 220 mm | 195 mm | 195 mm |
each line | |||
image volume | 512 × 512 × 400 | 512 × 512 × 161 | 512 × 512 × 161 |
dimension | |||
voxel size | (0.422 mm)3 | 0.48 × 0.48 × | 0.48 × 0.48 × |
0.50 mm3 | 0.50 mm3 | ||
Claims (7)
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US11/057,978 US7359477B2 (en) | 2005-02-15 | 2005-02-15 | Method for reconstructing a CT image using an algorithm for a short-scan circle combined with various lines |
CN200610009268.7A CN100528088C (en) | 2005-02-15 | 2006-02-15 | Method for reconstructing a ct image using an algorithm for a short-scan circle combined with various lines |
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US20080080758A1 (en) * | 2006-09-21 | 2008-04-03 | Stefan Hoppe | Method for determining final projection matrices |
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US20100034342A1 (en) * | 2008-08-07 | 2010-02-11 | Koninklijke Philips Electronics N. V. | Method and apparatus for correcting artifacts in circular ct scans |
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US20100286928A1 (en) * | 2009-05-08 | 2010-11-11 | Frank Dennerlein | Method and device for determining images from x-ray projections |
DE102012207910A1 (en) * | 2012-05-11 | 2013-11-14 | Siemens Aktiengesellschaft | Method for generating projection image sequence for reconstruction of object from image, involves generating trajectories for sources and detector for recording projections of object along trajectories, which comprise two portions |
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